BIT Numerical Mathematics

, Volume 2, Issue 4, pp 232–255 | Cite as

An arsenal of ALGOL procedures for complex arithmetic


This paper contains a complete system of ALGOL procedures which enable arithmetic operations to be carried out upon complex numbers. Further procedures for carrying out the evaluation of certain elementary functions (e.g. ln, exp, sin, ...) of a complex variable are given. Application of these procedures is then illustrated by their use in the computation of the confluent hypergeometric function and the Weber parabolic cylinder function. Procedures relating to the application of the ε-algorithm to series of complex terms are described. Two integrated series of procedures, relating to Stieltjes typeS-fractions and to corresponding continued fractions respectively, are given. Complete programmes, which illustrate the use of these procedures, may be used for the computation of the incompleteβ-function, the incompleteΓ-function (of arguments of large and small modulus) and the Weber function.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Backus, J. W. et al.,Report on the Algorithmic Language ALGOL 60, Num. Math., vol. 2, 1960, p. 106.Google Scholar
  2. 2.
    Fröberg, C. E.,Rational Chebyshev Approximation of Elementary Functions, BIT, vol. 1, 1961, p. 256.Google Scholar
  3. 3.
    Wynn, P.,On a Device for Computing the e m(Sn)Transformation, M.T.A.C., vo. 52, 1956, p. 663.Google Scholar
  4. 4.
    Szegö, G.,Über orthogonale Polynome, die zu einer gegebener Kurve der komplexen Ebene gehören, Math. Z., vol. 9, 1921, 0. 218.Google Scholar
  5. 5.
    Todd, J. and Warschawski, S. E.,On the Solution of the Lichtenstein-Gershgorin Integral Equation in Conformal Mapping: II Computational Experiments, N.B.S., Appl. Math. Ser. 42, p. 31.Google Scholar
  6. 6.
    Wynn, P.,The Rational Approximation of Functions which are Formally Defined by a Power Series Expansion, Maths. of Comp., vo. 14, 1960, p. 147.Google Scholar
  7. 7.
    Chebyshev, P.,Sur les Fractions Continues, Journ. de Math., vol. 8, 1858, p. 289.Google Scholar
  8. 8.
    Stieltjes, T. J.,Recherches sur les Fractions Continues, Annales de la Faculté des Sciences de Toulose, (1), 8, 1894, TI-122, 9, 1895, A5–57.Google Scholar
  9. 9.
    Markoff, A.,On Certain Applications of Algebraic Continued Fractions, Thesis, St. Petersburg, 1884.Google Scholar
  10. 10.
    Markoff, A.,Proof of the Convergence of Many Continued Fractions, Trans. Roy. Acad. Sci., St. Petersburg, 1893 (supp.).Google Scholar
  11. 11.
    Markoff, A.,On Functions Generated by Developing Power Series in Continued Fractions, Trans. Roy. Acad. Sci., St. Petersburg, 1894 (supp.).Google Scholar
  12. 12.
    Markoff, A.,Note sur les Fractions Continues, Bulletin de la Classe Physico-Mathématique de l'Académie Impériale des Sciences de Saint-Petersburg, vol. 5, 1895, p. 9.Google Scholar
  13. 13.
    Markoff, A.,Deux Demonstrations de la Convergence de Certaines Fractions Continues, Acta Mathematica, vol. 19, 1895, p. 93.Google Scholar
  14. 14.
    Markoff, A.,Nouvelles Applications des Fractions Continues, Memoires de l'Académie des Sciences de St. Petersburg, Classe Physico-Mathématique, vol. 3, 1896.Google Scholar
  15. 15.
    Markoff, A.,Nouvelles Applications des Fractions Continues, Math. Annalen, vol. 47, 1896, p. 579.Google Scholar
  16. 16.
    Shohat, J. A. and Tamarkin, J. D.,The Problem of Moments, Mathematical Surveys 1, Amer. Math. Soc. 1943.Google Scholar
  17. 17.
    Nevanlinna, R.,Asymptotische Entwicklungen beschränkter Funktionen und das Stieltjes Momenten Problem, Annales Academiae Fenniae, (A), vol. 18, 1922.Google Scholar
  18. 18.
    Carleman, T.,Les Fractions Quasi-analytiques, Gauthier-Villars, Paris 1926.Google Scholar
  19. 19.
    Rutishauser, H.,Der Quotienten-Differenzen Algorithmus, Birkhauser Verlag, Basel, 1957.Google Scholar
  20. 20.
    Goodwin, E. T. and Staton, J., Table of\(\int_0^\infty {e^{ - u^2 } du/(u + x)} \), Quart. Journ. Mech. and Appl. Math., vol. 1, 1948, p. 319.Google Scholar
  21. 21.
    Perron, O.,Die Lehre von den Kettenbrüchen, vol. II, Teubner, Stuttgart, 1957.Google Scholar
  22. 22.
    van Vleck, E. V.,On the Convergence of Algebraic Continued Fractions whose Coefficients have Limiting Values, Trans. Amer. Math. Soc., vol. 5, 1904, p. 253.Google Scholar
  23. 23.
    Ramanujan, S.,Collected Papers, Cambridge 1927.Google Scholar
  24. 24.
    Wynn, P.,The numerical Efficiency of Certain Continued Fraction Expansions, Proc. Kon. Ned. Akad. Wetensch. Amsterdam, vol. 65, ser. A, 1962, p. 127.Google Scholar
  25. 25.
    Wynn, P.,Numerical Efficiency Profile Functions, Proc. Kon. Ned. Akad. Wetensch. Amsterdam, vol. 65, ser. A, 1962, p. 118.Google Scholar
  26. 26.
    Erdélyi, A. et al.,Higher Transcendental Functions, McGraw-Hill.Google Scholar
  27. 27.
    Clenshaw, C. W.,Chebyshev Series for Mathematical Functions, Mathematical Tables, vol. 5, H.M.S.O., London 1962.Google Scholar

Copyright information

© BIT Foundations 1962

Authors and Affiliations

  • P. Wynn
    • 1
  1. 1.Mathematisch CentrumAmsterdam

Personalised recommendations